Abstract
We present a fast direct solver methodology for the Dirichlet biharmonic problem in a rectangle. The solver is applicable in the case of the second order Stephenson scheme [J. W. Stephenson, J. Comput. Phys., 55 (1984), pp. 65-80] as well as in the case of a new fourth order scheme, which is discussed in this paper and is based on the capacitance matrix method ([B. L. Buzbee and F. W. Dorr, SIAM J. Numer. Anal., 11 (1974), pp. 1136-1150], [P. Bjørstad, SIAM J. Numer. Anal., 20 (1983), pp. 59-71]). The discrete biharmonic operator is decomposed into two components. The first is a diagonal operator in the eigenfunction basis of the Laplacian, to which the FFT algorithm is applied. The second is a low-rank perturbation operator (given by the capacitance matrix), which is due to the deviation of the discrete operators from diagonal form. The Sherman-Morrison formula [G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed., John Hopkins University Press, Baltimore, MD, 1996] is applied to obtain a fast solution of the resulting linear system of equations.
Original language | English |
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Pages (from-to) | 303-333 |
Number of pages | 31 |
Journal | SIAM Journal on Scientific Computing |
Volume | 31 |
Issue number | 1 |
DOIs | |
State | Published - 2008 |
Keywords
- Biharmonic problem
- Capacitance matrix method
- Compact scheme
- Driven cavity
- Fast Fourier transform
- Fast solver
- Navier-Stokes equations
- Sherman-Morrison formula
- Stephenson scheme
- Streamfunction formulation
- Vorticity